The $10 gift card was for the delay, according to the email Amazon sent a while ago. The email where they said they will be sending the game first and then the watches when they are available. Sounds like the CSRs at Amazon are clueless like usual.
This is correct. If an Amazon rep tells you otherwise then they don't know what they're talking about. Shocking, I know. Here are the relevant parts of the email (mine is for the 3DS version):
"We're contacting you about your order for Zero Time Dilemma: Limited Watch Bonus Edition 3DS. We recently learned of an unexpected delay in getting the Limited Edition Bonus watch included in your order from our supplier. We’ll be shipping the game without the bonus watch and will ship the bonus watch as soon as it becomes available. We're very sorry about this.
You'll still receive the item and you can track the status of or make any changes to your order under Your Orders on Amazon.com...To help make up for the inconvenience, we've added a $10 promotional certificate to your account."
Of course, whether or not Amazon actually comes through remains to be seen. They've generally been a couple steps behind everyone else during this whole shitshow, but so far they've delivered, so I'm not too worried yet.